arXiv subjects
Publications and source records attributed to Xiaoyu He.
Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$, $$ (3\ell - 1) \cdot 2^{n-5} < ES_{\ell}(n) < \ell^2 \cdot 2^{n+ C\sqrt{n\log n}}.$$ A similar extension of the well-known Erd\H os--Szekeres cups-caps theorem is also proved.
A group of players are supposed to follow a prescribed profile of strategies. If they follow this profile, they will reach a given target. We show that if the target is not reached because some player deviates, then an outside observer can identify the deviator. We also construct identification methods in two nontrivial cases.
A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \ge 2^{Ω_H(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_n^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear.
Let $\mathcal{SS}_k(n)$ be the family of {\it shuffle squares} in $[k]^{2n}$, words that can be partitioned into two disjoint identical subsequences. Let $\mathcal{RSS}_k(n)$ be the family of {\it reverse shuffle squares} in $[k]^{2n}$, words that can be partitioned into two disjoint subsequences which are reverses of each other. Henshall, Rampersad, and Shallit conjectured asymptotic formulas for the sizes of $\mathcal{SS}_k(n)$ and $\mathcal{RSS}_k(n)$ based on numerical evidence. We prove that \[ \lvert \mathcal{SS}_k(n) \rvert=\dfrac{1}{n+1}\dbinom{2n}{n}k^n-\dbinom{2n-1}{n+1}k^{n-1}+O_n(k^{n-2}), \] confirming their conjecture for $\mathcal{SS}_k(n)$. We also prove a similar asymptotic formula for reverse shuffle squares that disproves their conjecture for $\lvert \mathcal{RSS}_k(n) \rvert$. As these asymptotic formulas are vacuously true when the alphabet size is small, we study the binary case separately and prove that $|\mathcal{SS}_2(n)| \ge \binom{2n}{n}$.
The Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$ vertices contains $G$ as a subgraph or its complement contains $H$ as a subgraph. For integers $n \geq k \geq 1$, the $k$-book $B_{k,n}$ is the graph on $n$ vertices consisting of a copy of $K_k$, called the spine, as well as $n-k$ additional vertices each adjacent to every vertex of the spine and non-adjacent to each other. A connected graph $H$ on $n$ vertices is called $p$-good if $r(K_p,H)=(p-1)(n-1)+1$. Nikiforov and Rousseau proved that if $n$ is sufficiently large in terms of $p$ and $k$, then $B_{k,n}$ is $p$-good. Their proof uses Szemerédi's regularity lemma and gives a tower-type bound on $n$. We give a short new proof that avoids using the regularity method and shows that every $B_{k,n}$ with $n \geq 2^{k^{10p}}$ is $p$-good. Using Szemerédi's regularity lemma, Nikiforov and Rousseau also proved much more general goodness-type results, proving a tight bound on $r(G,H)$ for several families of sparse graphs $G$ and $H$ as long as $|V(G)| < δ|V(H)|$ for a small constant $δ> 0$. Using our techniques, we prove a new result of this type, showing that $r(G,H) = (p-1)(n-1)+1$ when $H =B_{k,n}$ and $G$ is a complete $p$-partite graph whose first $p-1$ parts have constant size and whose last part has size $δn$, for some small constant $δ>0$. Again, our proof does not use the regularity method, and thus yields double-exponential bounds on $δ$.
The Longest Common Subsequence (LCS) is a fundamental string similarity measure, and computing the LCS of two strings is a classic algorithms question. A textbook dynamic programming algorithm gives an exact algorithm in quadratic time, and this is essentially best possible under plausible fine-grained complexity assumptions, so a natural problem is to find faster approximation algorithms. When the inputs are two binary strings, there is a simple $\frac{1}{2}$-approximation in linear time: compute the longest common all-0s or all-1s subsequence. It has been open whether a better approximation is possible even in truly subquadratic time. Rubinstein and Song showed that the answer is yes under the assumption that the two input strings have equal lengths. We settle the question, generalizing their result to unequal length strings, proving that, for any $\varepsilon>0$, there exists $δ>0$ and a $(\frac{1}{2}+δ)$-approximation algorithm for binary LCS that runs in $n^{1+\varepsilon}$ time. As a consequence of our result and a result of Akmal and Vassilevska-Williams, for any $\varepsilon>0$, there exists a $(\frac{1}{q}+δ)$-approximation for LCS over $q$-ary strings in $n^{1+\varepsilon}$ time. Our techniques build on the recent work of Guruswami, He, and Li who proved new bounds for error-correcting codes tolerating deletion errors. They prove a combinatorial "structure lemma" for strings which classifies them according to their oscillation patterns. We prove and use an algorithmic generalization of this structure lemma, which may be of independent interest.
We elucidate the relationship between the threshold and the expectation-threshold of a down-set. Qualitatively, our main result demonstrates that there exist down-sets with polynomial gaps between their thresholds and expectation-thresholds; in particular, the logarithmic gap predictions of Kahn--Kalai and Talagrand (recently proved by Park--Pham and Frankston--Kahn--Narayanan--Park) about up-sets do not apply to down-sets. Quantitatively, we show that any collection $\mathcal{G}$ of graphs on $[n]$ that covers the family of all triangle-free graphs on $[n]$ satisfies the inequality $\sum_{G \in \mathcal{G}} \exp(-δe(G^c) / \sqrt{n}) < 1/2$ for some universal $δ> 0$, and this is essentially best-possible.
A CNOT circuit is the key gadget for entangling qubits in quantum computing systems. However, the qubit connectivity of noisy intermediate-scale quantum (NISQ) devices is constrained by their {limited connectivity architecture}. To improve the performance of CNOT circuits on NISQ devices, we investigate the optimization of the size/depth of CNOT circuits under the limited connectivity architecture. We present a method that can optimize the size of any $n$-qubit CNOT circuit $O\left(\frac{n^2}{\log δ}\right)$ on any connected graph with minimum degree $δ$, and prove this bound is optimal for the regular graph. For the near-term sparsely connected structure, we additionally present a method that can optimize the size of any $n$-qubit CNOT circuit to below $2n^2$. The numerical experiment shows that our method performs better than state-of-the-art results. Specifically, we present an example to illustrate the applicability of our algorithm. For the grid structure, which is commonly used in current quantum devices, we demonstrate that the depth of any $n$-qubit CNOT circuit can be optimized to be linear in $n$ with certain ancillary qubits (ancillas). Experimental results indicate that this method has significant improvements compared with all of the existing methods. We additionally test our algorithms on the five-qubit IBMQ devices, and the experiments show that the measurement results of the optimized circuit with our algorithm are more robust to noise compared with the IBM mapping method.
A family of sets $A$ is said to be an antichain if $x\not\subset y$ for all distinct $x,y\in A$, and it is said to be a distance-$r$ code if every pair of distinct elements of $A$ has Hamming distance at least $r$. Here, we prove that if $A\subset 2^{[n]}$ is both an antichain and a distance-$(2r+1)$ code, then $|A| = O_r(2^n n^{-r-1/2})$. This result, which is best-possible up to the implied constant, is a purely combinatorial strengthening of a number of results in Littlewood--Offord theory; for example, our result gives a short combinatorial proof of Hálasz's theorem, while all previously known proofs of this result are Fourier-analytic.
We prove that there exists an absolute constant $δ>0$ such any binary code $C\subset\{0,1\}^N$ tolerating $(1/2-δ)N$ adversarial deletions must satisfy $|C|\le 2^{\text{poly}\log N}$ and thus have rate asymptotically approaching 0. This is the first constant fraction improvement over the trivial bound that codes tolerating $N/2$ adversarial deletions must have rate going to 0 asymptotically. Equivalently, we show that there exists absolute constants $A$ and $δ>0$ such that any set $C\subset\{0,1\}^N$ of $2^{\log^A N}$ binary strings must contain two strings $c$ and $c'$ whose longest common subsequence has length at least $(1/2+δ)N$. As an immediate corollary, we show that $q$-ary codes tolerating a fraction $1-(1+2δ)/q$ of adversarial deletions must also have rate approaching 0. Our techniques include string regularity arguments and a structural lemma that classifies binary strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence.
Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number $r(K_{4}^{(3)},S_n^{(3)})$ exhibits an unusual intermediate growth rate, namely, \[ 2^{c \log^2 n} \le r(K_{4}^{(3)},S_n^{(3)}) \le 2^{c' n^{2/3}\log n} \] for some positive constants $c$ and $c'$. The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum $N$ such that any $2$-edge-coloring of the Cartesian product $K_N \square K_N$ contains either a red rectangle or a blue $K_n$?
Federated learning allows multiple participants to collaboratively train an efficient model without exposing data privacy. However, this distributed machine learning training method is prone to attacks from Byzantine clients, which interfere with the training of the global model by modifying the model or uploading the false gradient. In this paper, we propose a novel serverless federated learning framework Committee Mechanism based Federated Learning (CMFL), which can ensure the robustness of the algorithm with convergence guarantee. In CMFL, a committee system is set up to screen the uploaded local gradients. The committee system selects the local gradients rated by the elected members for the aggregation procedure through the selection strategy, and replaces the committee member through the election strategy. Based on the different considerations of model performance and defense, two opposite selection strategies are designed for the sake of both accuracy and robustness. Extensive experiments illustrate that CMFL achieves faster convergence and better accuracy than the typical Federated Learning, in the meanwhile obtaining better robustness than the traditional Byzantine-tolerant algorithms, in the manner of a decentralized approach. In addition, we theoretically analyze and prove the convergence of CMFL under different election and selection strategies, which coincides with the experimental results.
Consider the following experiment: a deck with $m$ copies of $n$ different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of "feedback" is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in $n$, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.
For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed to be a monochromatic clique on $n$ vertices, that is, a subset of $n$ vertices where all of the edges between them receive a common color. In particular, the case $s=1$ corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on $R(n;r,s)$ which imply that $R(n;r,s) = 2^{Θ(nr)}$ if $s/r$ is bounded away from $0$ and $1$. The upper bound extends an old result of Erdős and Szemerédi, who treated the case $s = r-1$, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.
This work concerns the evolutionary approaches to distributed stochastic black-box optimization, in which each worker can individually solve an approximation of the problem with nature-inspired algorithms. We propose a distributed evolution strategy (DES) algorithm grounded on a proper modification to evolution strategies, a family of classic evolutionary algorithms, as well as a careful combination with existing distributed frameworks. On smooth and nonconvex landscapes, DES has a convergence rate competitive to existing zeroth-order methods, and can exploit the sparsity, if applicable, to match the rate of first-order methods. The DES method uses a Gaussian probability model to guide the search and avoids the numerical issue resulted from finite-difference techniques in existing zeroth-order methods. The DES method is also fully adaptive to the problem landscape, as its convergence is guaranteed with any parameter setting. We further propose two alternative sampling schemes which significantly improve the sampling efficiency while leading to similar performance. Simulation studies on several machine learning problems suggest that the proposed methods show much promise in reducing the convergence time and improving the robustness to parameter settings.
This work provides an efficient sampling method for the covariance matrix adaptation evolution strategy (CMA-ES) in large-scale settings. In contract to the Gaussian sampling in CMA-ES, the proposed method generates mutation vectors from a mixture model, which facilitates exploiting the rich variable correlations of the problem landscape within a limited time budget. We analyze the probability distribution of this mixture model and show that it approximates the Gaussian distribution of CMA-ES with a controllable accuracy. We use this sampling method, coupled with a novel method for mutation strength adaptation, to formulate the mixture model based evolution strategy (MMES) -- a CMA-ES variant for large-scale optimization. The numerical simulations show that, while significantly reducing the time complexity of CMA-ES, MMES preserves the rotational invariance, is scalable to high dimensional problems, and is competitive against the state-of-the-arts in performing global optimization.
The list Ramsey number $R_{\ell}(H,k)$, recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list-coloring variant of the classical Ramsey number. They showed that if $H$ is a fixed $r$-uniform hypergraph that is not $r$-partite and the number of colors $k$ goes to infinity, $e^{Ω(\sqrt{k})} \le R_{\ell} (H,k) \le e^{O(k)}$. We prove that $R_{\ell}(H,k) = e^{Θ(k)}$ if and only if $H$ is not $r$-partite.
Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr--Erdős conjecture, answering a question of Bucić, Letzter, and Sudakov. If $H$ is an acyclic digraph, the oriented Ramsey number of $H$, denoted $\overrightarrow{r_{1}}(H)$, is the least $N$ such that every tournament on $N$ vertices contains a copy of $H$. We show that for any $Δ\geq 2$ and any sufficiently large $n$, there exists an acyclic digraph $H$ with $n$ vertices and maximum degree $Δ$ such that \[ \overrightarrow{r_{1}}(H)\ge n^{Ω(Δ^{2/3}/ \log^{5/3} Δ)}. \] This proves that $\overrightarrow{r_{1}}(H)$ is not always linear in the number of vertices for bounded-degree $H$. On the other hand, we show that $\overrightarrow{r_{1}}(H)$ is nearly linear in the number of vertices for typical bounded-degree acyclic digraphs $H$, and obtain linear or nearly linear bounds for several natural families of bounded-degree acyclic digraphs. For multiple colors, we prove a quasi-polynomial upper bound $\overrightarrow{r_{k}}(H)=2^{(\log n)^{O_{k}(1)}}$ for all bounded-degree acyclic digraphs $H$ on $n$ vertices, where $\overrightarrow{r_k}(H)$ is the least $N$ such that every $k$-edge-colored tournament on $N$ vertices contains a monochromatic copy of $H$. For $k\geq 2$ and $n\geq 4$, we exhibit an acyclic digraph $H$ with $n$ vertices and maximum degree $3$ such that $\overrightarrow{r_{k}}(H)\ge n^{Ω(\log n/\log\log n)}$, showing that these Ramsey numbers can grow faster than any polynomial in the number of vertices.